The unique equilibrium’s global exponential asymptotic stability of the irreversible Michaelis–Menten mechanism of enzyme kinetics

In this work, we investigate the dynamical system of irreversible Michaelis–Menten kinetics in enzyme kinetics. We first summarize some important theoretical results such as non-negativity, boundedness or conservation laws for solutions of the time-continuous dynamical system. As our main contribution regarding this time-continuous setting, we demonstrate that the unique equilibrium state is globally exponentially asymptotically stable by providing a suitable Lyapunov-function. Based on the implicit Eulerian time-stepping method, we propose an explicit reformulation of this time-discretization for the numerical simulation. Our main result for the time-discrete dynamical system is that the unique equilibrium state of the time-discretization is globally exponentially asymptotically stable as well. Conclusively, we illustrate our theoretical findings by numerical experiments.